On a profit maximizing location model

In this paper we discuss a locational model with a profit-maximizing objective. The model can be illustrated by the following situation. There is a set of potential customers in a given region. A firm enters the market and wants to sell a certain product to this set of customers. The location and demand of each potential customer are assumed to be known. In order to maximize its total profit, the firm has to decide: (1) where to locate its distribution warehouse to serve the customers; (2) the price for its product. Due to existence of competition, each customer holds a reservation price for the product. This reservation price is a decreasing function of the distance to the warehouse. If the actual price is higher than the reservation price, then the customer will turn to some other supplier and hence is lost from the firm's market. The problem of the firm is to find the best location for its warehouse and the best price for its product at the same time in order to maximize the total profit. We show that under certain assumptions on the complexity counts, a special case of this problem can be solved in polynomial time.